91Ó°ÊÓ

A parabolic cylinder is a three-dimensional shape created by extending a parabola along a line, often the x-axis, to form a surface. Unlike a standard cylinder, which is based on a circle, a parabolic cylinder takes its cross-sectional shape from a parabola. In our exercise, the parabolic cylinder is described by the equation \( y^2 + 4z = 16 \). This specific equation gives us all the points \((y, z)\) that lie on the surface for any fixed value of \(x\).
The equation can be rearranged to solve for \(z\), showing the relationship between \(y\) and \(z\):

• When \(y\) changes, \(z\) adjusts according to \(z = \frac{16-y^2}{4}\).
• The surface extends infinitely in one direction but is bounded by given limits in this exercise.

In practical terms, the parabolic cylinder in our task is truncated by planes, making it easier to visualize and integrate over. You can imagine it as a curved wall within specified boundaries.

Triple integration involves calculating the integral of a function in three dimensions, usually to determine the volume under a surface or the total of some property, like mass or charge, contained within a 3D space. In the context of this problem, we integrate a function \(g(x, y, z) = x \sqrt{y^2 + 4}\) over a defined region bounded by a parabolic cylinder and planes.
The process involves several steps:

• First, integrating with respect to one variable, often starting with \(z\), because it is the innermost integration.
• Then integrating with respect to another variable, \(y\), while keeping \(x\) constant.
• Finally, integrating with respect to \(x\).

Each integration step narrows down the dimensions of our problem until we finally reach a single numerical answer, representing the total within that region. Triple integration requires careful setup of limits, ensuring every dimension is accounted for correctly.

When setting up an integral, limits of integration are essential for defining the region over which you integrate. These limits tell you where integration starts and stops for each variable. In some cases, these are constants, but in our problem, certain limits are functions of other variables.
For the given surface bounded by a parabolic cylinder, the limits were:

• For \(x\), the limits are simple constants from 0 to 1, representing the portion of the x-axis.
• For \(y\), the limits range from -4 to 4, which are found by examining where the parabolic cross-section intersects with the plane \(z=0\), when \(y^2 = 16\).
• For \(z\), the upper limit changes with \(y\): \(z = \frac{16 - y^2}{4}\). This shows that as \(y\) varies, the height of the cylinder at that point changes.

Understanding these limits is crucial for accurately setting up and computing the triple integral. They define the shape and size of the region you are integrating over, ensuring you capture the entire volume enclosed by the boundaries.